New Syllabus 2027
Form Three Basic Mathematics Notes
Chapter One: Relations · Ordinary Level (Form Three) · Study Notes. Read this chapter free, then continue every chapter in the Click Online Academy app.
All Form Three Basic Mathematics Chapters
- Chapter 1RelationsRead below · Free
- Chapter 2FunctionsContinue in the app
- Chapter 3Linear ProgrammingContinue in the app
- Chapter 4Sequence and SeriesContinue in the app
- Chapter 5CirclesContinue in the app
Learning Outcomes
By the end of this chapter, you should be able to:
- explain the meaning of a relation and represent it using ordered pairs, an arrow diagram, and a table;
- identify and classify a relation as one-to-one, one-to-many, many-to-one, or many-to-many;
- state and illustrate the domain and range of a relation;
- draw and interpret graphs of relations given by linear equations, linear inequalities, and quadratic equations;
- determine the domain and range of a relation expressed in equation form;
- find the inverse of a relation and sketch the graph of an inverse relation.
Key Terms and Concepts
| Term | Meaning |
|---|---|
| Set | A well-defined collection of distinct objects (elements). Named with capital letters, e.g. A = {1, 2, 3}. |
| Ordered pair | A pair written in fixed order (x, y). Order matters: (2, 5) ≠ (5, 2). |
| Cartesian product | Given sets A and B, A × B is the set of all ordered pairs (a, b) with a ∈ A and b ∈ B. |
| Relation | A relation from set A to set B is any subset of A × B — a rule that connects some or all elements of A to some or all of B. |
| Domain | The set of all first components (inputs) actually used in the relation. |
| Range | The set of all second components (outputs) that are actually used. |
| Inverse relation | The relation obtained by interchanging the components of every ordered pair, denoted R⁻¹. |
1. Meaning of a Relation
In everyday life we connect two groups of things: pupils to their class streams, countries to their capital cities, or numbers to their squares. In mathematics such a connection is called a relation.
Ways of representing a relation
- Set of ordered pairs: e.g. R = {(1,2), (2,4), (3,6)}
- Arrow (mapping) diagram: arrows from elements of the domain to elements of the range
- Table of values
- Equation: e.g. y = 2x
- Graph on the Cartesian plane
TEST YOUR UNDERSTANDING
Q1. What is a relation from set A to set B?
Answer: Any subset of the Cartesian product A × B; a rule that connects some or all elements of A to some or all of B.
Q2. Given A = {3, 5} and B = {2, 4, 6}, list all elements of A × B and state n(A × B).
Answer: A × B = {(3,2), (3,4), (3,6), (5,2), (5,4), (5,6)}. n(A × B) = 2 × 3 = 6.
2. Types of Relations
Relations are classified according to how elements of the domain are paired with elements of the range:
| Type | Description | Example |
|---|---|---|
| One-to-one | Each element of the domain maps to exactly one unique element of the range, and no two domain elements share the same range element. | {(1,2), (2,4), (3,6)} |
| One-to-many | At least one element of the domain maps to two or more elements of the range. | {(2,3), (2,5)} |
| Many-to-one | Two or more elements of the domain map to the same element of the range. | {(5,1), (6,1), (7,2)} |
| Many-to-many | At least one domain element maps to several range elements, and at least one range element is shared by several domain elements. | Friendship among a group of people |
TEST YOUR UNDERSTANDING
Q3. State the type of relation represented by R = {(5,1), (6,1), (7,2)}.
Answer: Many-to-one (5 and 6 both map to 1).
3. Domain and Range of a Relation
- The domain is the set of all first components (x-values / inputs) that appear in the relation.
- The range is the set of all second components (y-values / outputs) that appear in the relation.
- The co-domain is the whole target set; the range is always a subset of the co-domain.
Domain and range from an equation
- Exclude values that cause division by zero or a negative number under a square root (for real numbers).
- Example: for y = 1/(x + 6), domain is all real x except x = −6.
- Example: for y = √(x − 5), domain is x ≥ 5; range is y ≥ 0.
TEST YOUR UNDERSTANDING
Q4. Find the domain and range of R = {(-1,2), (0,4), (1,2), (2,8)}.
Answer: Domain = {−1, 0, 1, 2}. Range = {2, 4, 8}.
Q5. Find the domain of y = 1/(x + 6).
Answer: x ≠ −6, so Dom = {x : x ∈ ℝ, x ≠ −6}.
4. Graphs of Relations
- Linear equation (e.g. y = 2x + 1) → straight line.
- Linear inequality (e.g. y ≤ 2x + 1) → region shaded on one side of a solid (≤, ≥) or broken (<, >) boundary line.
- Quadratic equation (e.g. y = x² − 1) → parabola.
5. Inverse of a Relation
- Domain of R⁻¹ = Range of R
- Range of R⁻¹ = Domain of R
- The graph of R⁻¹ is the reflection of the graph of R in the line y = x.
TEST YOUR UNDERSTANDING
Q6. Find the inverse of R = {(2,6), (3,9)}.
Answer: R⁻¹ = {(6,2), (9,3)}.
Q7. Find the inverse of y = 4x − 3.
Answer: Swap: x = 4y − 3. Solve: 4y = x + 3, so y = (x + 3)/4.
Q8. In which line is the graph of a relation reflected to obtain the graph of its inverse?
Answer: The line y = x.
Chapter Summary
- A relation from set A to set B is any subset of A × B; it can be shown in words, as ordered pairs, as an arrow diagram, or as an equation.
- There are four types of relations: one-to-one, one-to-many, many-to-one, and many-to-many.
- The domain is the set of first components used; the range is the set of second components used.
- Relations given by equations can be graphed: linear equations give straight lines, inequalities give shaded regions, quadratics give parabolas.
- The Vertical Line Test tells whether a graphed relation is a function.
- The inverse R⁻¹ is obtained by swapping components (or interchanging x and y in an equation). Its graph is the reflection of the graph of R in the line y = x.
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Continue in the appEnd-of-Topic Questions
Attempt all questions in your exercise book before checking the suggested answers. (New Syllabus 2027)
- Given A = {3, 5} and B = {2, 4, 6}, list all the elements of A × B and state n(A × B). (3 marks)
- Write the relation R: “x divides y” as a set of ordered pairs, where A = {2, 3} and B = {6, 9, 12}. (3 marks)
- State the type of relation represented by R = {(5,1), (6,1), (7,2)}. (2 marks)
- Give one real-life example each of a one-to-one relation and a many-to-many relation. (4 marks)
- Find the domain and range of R = {(-1,2), (0,4), (1,2), (2,8)}. (3 marks)
- Find the domain of the relation y = 1/(x + 6). (2 marks)
- Find the domain and range of y = √(x − 5). (3 marks)
- Find the inverse of the relation y = 4x − 3. (3 marks)
- If R = {(2,9), (3,11)}, find R⁻¹ and state its domain and range. (4 marks)
- State the Vertical Line Test and use it to explain whether the graph of x = y² represents a function. (4 marks)
- A relation R has domain {1, 2, 3, 4} and every element maps to the single value 10. State the type of relation and write R as a set of ordered pairs. (3 marks)
- Describe how the graph of a relation and the graph of its inverse are related. (3 marks)
Suggested Answers
- A × B = {(3,2), (3,4), (3,6), (5,2), (5,4), (5,6)}. n(A × B) = 6.
- R = {(2,6), (2,12), (3,6), (3,9), (3,12)}.
- Many-to-one.
- One-to-one: “is the national ID number of”. Many-to-many: “is a friend of” among a group of people.
- Domain = {−1, 0, 1, 2}. Range = {2, 4, 8}.
- x ≠ −6, so Dom = {x : x ∈ ℝ, x ≠ −6}.
- Domain: x ≥ 5. Range: y ≥ 0.
- y = (x + 3)/4.
- R⁻¹ = {(9,2), (11,3)}. Domain of R⁻¹ = {9, 11}; Range of R⁻¹ = {2, 3}.
- A relation is a function only if every vertical line crosses its graph at most once. For x = y² a vertical line such as x = 4 crosses at both y = 2 and y = −2, so it is not a function.
- Many-to-one. R = {(1,10), (2,10), (3,10), (4,10)}.
- The graph of the inverse is the reflection (mirror image) of the graph of the original relation in the line y = x.
These are suggested answers. Credit is also given to correct points expressed in your own words.
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